Fractional Commutative Hyperoperators: Investigation

Generated by Claude (Sonnet 4.5) on 2025-10-13

Motivation

We want to find whether commutative hyperoperators can be extended to fractional ranks (e.g., operation 2.5, halfway between multiplication and commutative exponentiation) that would:

1. Complete the "triangle of power" at each level
2. Allow fractional operations like ⊕ₙ^(1/k)(x) analogous to nth roots
3. Enable mean relationships at all levels analogous to arithmetic ↔ geometric mean

Background: What Exists

Fractional Iteration Theory

Schröder's Equation provides a framework for fractional iteration:

• For function f with fixed point and multiplier s
• Solve: Ψ(f(x)) = s·Ψ(x)
• Then fractional iterates: f^(t)(x) = Ψ^(-1)(st · Ψ(x))

Abel's Equation: α(f(x)) = α(x) + 1

• Related to Schröder via: Ψ(x) = s^(α(x))
• Also gives fractional iteration

Existing Work on Fractional Hyperoperators

• Fractional tetration has been studied extensively
• Extension to real/complex numbers was proved in 2017
• Active research area in functional equations
• Stack Exchange discussion mentions using "Schröder-mechanism" for fractional iteration heights with commutative hyperoperators

Ghalimi's Construction

The commutative hyperoperators are defined:

⊕₀(x, y) = log(exp(x) + exp(y))
⊕₁(x, y) = x + y
⊕₂(x, y) = x × y
⊕₃(x, y) = exp(log(x) × log(y))
⊕ₙ(x, y) = exp(⊕ₙ₋₁(log(x), log(y)))  [for n > 3]

Key property we proved: ln(⊕ₙ(x, y)) = ⊕ₙ₋₁(ln(x), ln(y))

Open question from Ghalimi: Can we define ⊕ₙ for n ∈ ℝ or ℂ?

Proposed Approaches

Approach 1: Direct Interpolation

Define ⊕_{n+t} for 0 < t < 1 using a smooth interpolation that satisfies:

1. Continuity: ⊕ₙ should vary continuously with n
2. Recursion: ⊕_{n+1}(x, y) = exp(⊕ₙ(log(x), log(y)))
3. Commutativity: ⊕{n+t}(x, y) = ⊕{n+t}(y, x)

Naive attempt: Linear interpolation in the transform space?

⊕_{n+t}(x, y) = exp((1-t)·⊕_{n-1}(log(x), log(y)) + t·⊕ₙ(log(x), log(y)))

Problem: This probably doesn't preserve the recursive structure or other properties.

Approach 2: Via Schröder's Equation

For a fixed y, consider the function:

fᵧ(x) = ⊕ₙ(x, y)

Try to solve Schröder's equation for this function, then use that to define fractional operations.

Challenge: The multiplier and fixed point structure may not be straightforward.

Approach 3: Fractional Composition

The recursive structure suggests:

⊕ₙ₊₁ = exp ∘ ⊕ₙ ∘ (log × log)

Can we define "fractional composition" of the exp and log operations?

If we could define exp^(t) for 0 < t < 1 (fractional iteration of exp), then:

⊕_{n+t} = exp^(t) ∘ ⊕ₙ ∘ (log × log)^(t)

But: Fractional iteration of exp is notoriously difficult because exp has no fixed point in ℝ.

Approach 4: Power Series Extension

For small arguments, expand ⊕ₙ(x, y) as a power series in log(x) and log(y), then interpolate the coefficients smoothly as functions of n.

Challenge: Ensuring convergence and preserving properties.

What We Need for the "Triangle of Power"

At each level n ≥ 3, we want:

          ⊕ₙ(x, y)
          /        \
         /          \
    ln(·)  ←→  ⊕ₙ^(1/k)(x)

Where ⊕ₙ^(1/k) is a kth "root" operation satisfying:

⊕ₙ(⊕ₙ^(1/k)(x), ..., ⊕ₙ^(1/k)(x)) = x  [k times]

Critical property needed:

ln(⊕ₙ^(1/k)(x)) = (1/k) · something related to ln(x)

This would allow the mean relationship:

exp(generalized_mean_{n-1}(ln(odds))) = generalized_mean_n(odds)

Specific Test Cases

Case 1: ⊕_{2.5} (between multiplication and commutative exp)

We want ⊕_{2.5} such that:

• ⊕{2.5}(⊕{2.5}(x, y), ⊕_{2.5}(x, y)) ≈ ⊕₃(x, y)
• ⊕_{2.5}(x, y) is between x×y and exp(log(x)×log(y))
• ln(⊕{2.5}(x, y)) = ⊕{1.5}(ln(x), ln(y))

Computational test: For x=2, y=3:

• ⊕₂(2, 3) = 6
• ⊕₃(2, 3) = exp(log(2)×log(3)) ≈ 1.933
• ⊕_{2.5}(2, 3) should be ≈ 3.29 (geometric mean of 6 and 1.933)

Case 2: Square Root Operation

For ⊕₃ (commutative exponentiation), what is ⊕₃^(1/2)?

We want: ⊕₃(⊕₃^(1/2)(x), ⊕₃^(1/2)(x)) = x

So: exp(log(y)×log(y)) = x where y = ⊕₃^(1/2)(x) Therefore: y = exp(√(log(x)/log(y)))

This is an implicit equation! We'd need to solve for y.

Open Questions

1. Existence: Do fractional commutative hyperoperators exist that satisfy all desired properties?

2. Uniqueness: If they exist, are they unique? (Fractional iteration often has non-unique solutions)

3. Analyticity: Can they be extended to a holomorphic function of n?

4. Computation: Can they be computed efficiently? (Might require numerical solution of functional equations)

5. Triangle structure: Do fractional operations automatically give us the "third vertex" of the triangle at each level?

6. Mean relationships: If we can define fractional operations, do we automatically get generalized mean relationships?

Plausible but Unproven Conjecture

Conjecture: There exists a family of operations ⊕ₙ for n ∈ ℝ, n ≥ 0, such that:

1. ⊕ₙ₊₁(x, y) = exp(⊕ₙ(log(x), log(y))) for all n
2. ⊕ₙ is continuous in n
3. ⊕ₙ is commutative and associative for all n
4. For integer n, ⊕ₙ agrees with Ghalimi's construction

If true, this would enable a "tower of triangles" with mean relationships at every level.

But: Proving existence and constructing it explicitly are major open problems.

Where to Look Next

1. Tetration forums and research communities studying fractional tetration
2. Papers on functional equations - especially those using Schröder/Abel methods
3. Numerical methods - try to compute ⊕_{2.5} numerically and see if it has nice properties
4. Complex analysis - analytic continuation might be the right framework
5. Ghalimi himself - the Observable notebooks might have more details or he might be interested in this question

Relation to Original Question

This investigation was sparked by observing:

• The "triangle of power" (exp, log, √) at the exponentiation level
• Distributivity: log(a×b) = log(a) + log(b)
• These together enable: exp(arithmean(log(odds))) = geommean(odds)

For this pattern to extend to all levels with commutative hyperoperators, we need both:

1. ✓ Distributivity at all levels (we proved this)
2. ? Fractional operations to complete the triangles (still open)

Status: Partially Solved

• What exists: Theory of fractional iteration (Schröder, Abel)
• What's proven: Fractional tetration exists (2017)
• What's unclear: Whether Ghalimi's construction extends smoothly to fractional ranks
• What's missing: Explicit formulas and computation methods
• What's really missing: The connection to generalized means at each level

This appears to be novel mathematics if properly worked out, though the ingredients exist in the literature. The specific question of whether fractional commutative hyperoperators complete the "triangle of power" at each level seems unexplored.

Recommendation

This deserves a proper mathematical investigation, possibly:

1. A computational exploration (try to construct ⊕_{2.5} numerically)
2. An analytical approach using functional equations
3. Reaching out to researchers in fractional iteration / tetration

If this works out, it would be a beautiful unification of:

• Generalized means (Hölder)
• Commutative hyperoperators (Ghalimi)
• Fractional iteration (Schröder/Abel)
• The arithmetic-geometric mean connection

Update: Clarifying Two Distinct Problems

Added 2025-10-13 after further discussion

We realized there was confusion between two different notions of "fractional":

Problem 1: Fractional Hyperoperator Ranks

Finding operations ⊕{n+t} for non-integer n+t (like ⊕{2.5}, halfway between multiplication and commutative exponentiation).

Status: Theoretically possible via Schröder/Abel methods, no explicit formulas known for Ghalimi's construction.

Problem 2: Root Operations at Each Level

For each integer level n, finding an operation ⊕ₙ^(1/k) that acts like an "nth root" does for exponentiation.

Key requirement: ln(⊕ₙ^(1/k)(x)) should relate nicely to ln(x), allowing mean calculations.

Status: Unknown whether this is even possible.

The Triangle Structure

At the exponentiation level (n=2), we have a triangle of power:

        exp(x)
       /      \
      /        \
   log(x)  ←→  x^(1/n)

With the crucial property: log(x^(1/n)) = (1/n)·log(x)

This property enables: exp(arithmetic_mean(log(odds))) = geometric_mean(odds)

For higher levels, we'd need analogous triangles:

          ⊕ₙ(x, y)
          /        \
         /          \
    ln(·)  ←→  ⊕ₙ^(1/k)(·)

Two Paths Forward

Path A: Via Fractional Ranks

• Define ⊕_{n+t} for all real n
• Hope that this automatically gives us root operations ⊕ₙ^(1/k)
• Challenge: No explicit construction known

Path B: Define Roots Directly

• At each integer level n, separately define what ⊕ₙ^(1/k) means
• Don't require fractional ranks between levels
• Challenge: Unclear if this is possible with current structure

The Complex Number Obstacle

For n ≥ 4, commutative hyperoperators enter the complex domain:

• ln(ln(2)) < 0, so we need complex numbers
• Complex logarithms are multi-valued (branch cuts)
• Complex roots are multi-valued (n different nth roots)
• The triangle relationships might only hold for specific branch choices

Possible simplification: Restrict to positive reals x, y > e?

For x, y > e:

• ⊕₃(x, y) stays real and positive
• ⊕₄(x, y) = exp(⊕₃(ln(x), ln(y))) might stay real
• Need to verify this computationally

What We Actually Know

Proven:

1. Distributivity holds at all levels: ln(⊕ₙ(x, y)) = ⊕ₙ₋₁(ln(x), ln(y))
2. This works even in ℂ (by construction)

Unknown:

1. Whether fractional ranks exist with nice properties
2. Whether root operations ⊕ₙ^(1/k) can be defined
3. Whether these would complete the triangle structure
4. Whether this enables mean relationships at higher levels

The Deep Question

The arithmetic-geometric mean relationship works because of two independent structures:

1. Distributivity: log(a×b) = log(a) + log(b)
2. Triangle: log, exp, and roots interrelate via log(x^(1/n)) = (1/n)·log(x)

We have (1) at all levels for commutative hyperoperators.

Open question: Can we construct (2) at all levels?

This might require:

• Novel mathematics
• Numerical experimentation
• Better understanding of fractional iteration
• Or it might simply be impossible

Next Steps for Investigation

1. Computational experiments:
   • Try to numerically construct ⊕_{2.5}
   • Test whether it has nice properties
   • Check if it gives us root operations
2. Restrict to reals:
   • Work out the structure for x, y ∈ ℝ₊ first
   • Only extend to ℂ if the real case works
3. Contact experts:
   • Tetration forum researchers
   • Ismael Ghalimi (creator of the construction)
   • Functional equations specialists
4. Alternative approaches:
   • Maybe the triangle structure requires different hyperoperators?
   • Could there be a different construction that naturally includes roots?

Current Assessment

Likelihood of success: Uncertain but worth exploring

Potential impact: High - would unify several mathematical structures

Difficulty: Significant - requires solving open problems in fractional iteration

Time investment: Weeks to months of serious mathematical work

This is a promising research direction but not a quick win.